Certification cost of quantum models: measurement correlation, not parameter count
Abstract
Reporting the Fisher geometry of a trained variational quantum model is routine; quoting the shot budget that would establish it is not. Certifying an empirical Fisher matrix to relative Frobenius error ε under coordinate-wise parameter shift costs Θ(B p2 V/(ε2 G)) circuit executions, where V is the measured readout variance and G the measured squared gradient norm, with uniform allocation optimal in that class. One constant reproduces the cost of two circuit families whose exponents differ by a full power of p. The exponent is an identity in how nV and nG scale with the register, holding family by family to $0.001$ across 624 matrix-product-state cells once the finite-p prefactor is removed. The cubic cost is therefore a finite-size window, set by whether the readout light cone grows with the register. A product family to 256 qubits gives $1.966$ (95% CI $1.934−−1.997$); a brickwork entangler falls from $2.853$ below ten qubits to $1.715$ beyond sixty-four; a blocked entangler gives $1.984$ at a fixed cone width against $3.034$ at a proportional one. Fixed device connectivity fixes the cone, so a cubic budget from a small simulation overestimates a large machine, on top of hardware multipliers 2.07× ($1.41−−3.02$), 2.38× and 1.91× on ibm_marrakesh, ibm_fez and ibm_kingston. Cost-optimal readout weights cut the measured shot budget by 2.67× ($1.33−−4.00$) on hardware, flat from four to twelve qubits. A discrepancy model fitted on cheap circuits transfers its mean inside the calibration grid and, at six larger sizes named before the data, does not: nominal 90% intervals cover 36%, and split conformal is the only rung that stays near nominal.
Related
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BibTeX
@misc{sulimov2026certification,
title = {{Certification cost of quantum models: measurement correlation, not parameter count}},
author = {Pavel Sulimov and Claude Lehmann},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14424},
eprint = {2609.14424},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14424},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Certification cost of quantum models: measurement correlation, not parameter count
AU - Pavel Sulimov
AU - Claude Lehmann
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14424
UR - https://arxiv.org/abs/2609.14424
AB - Reporting the Fisher geometry of a trained variational quantum model is routine; quoting the shot budget that would establish it is not. Certifying an empirical Fisher matrix to relative Frobenius error ε under coordinate-wise parameter shift costs Θ(B p^2 V/(ε^2 G)) circuit executions, where V is the measured readout variance and G the measured squared gradient norm, with uniform allocation optimal in that class. One constant reproduces the cost of two circuit families whose exponents differ by a full power of p. The exponent is an identity in how nV and nG scale with the register, holding family by family to $0.001$ across 624 matrix-product-state cells once the finite-p prefactor is removed. The cubic cost is therefore a finite-size window, set by whether the readout light cone grows with the register. A product family to 256 qubits gives $1.966$ (95% CI $1.934−−1.997$); a brickwork entangler falls from $2.853$ below ten qubits to $1.715$ beyond sixty-four; a blocked entangler gives $1.984$ at a fixed cone width against $3.034$ at a proportional one. Fixed device connectivity fixes the cone, so a cubic budget from a small simulation overestimates a large machine, on top of hardware multipliers 2.07× ($1.41−−3.02$), 2.38× and 1.91× on ibm_marrakesh, ibm_fez and ibm_kingston. Cost-optimal readout weights cut the measured shot budget by 2.67× ($1.33−−4.00$) on hardware, flat from four to twelve qubits. A discrepancy model fitted on cheap circuits transfers its mean inside the calibration grid and, at six larger sizes named before the data, does not: nominal 90% intervals cover 36%, and split conformal is the only rung that stays near nominal.
ER -